# Valuation of Cash-Or-Nothing option

Studying options pricing, I'm stuck with the following problem:

The price of a stock is described by the dynamic: $$dS_t = \mu\, dt + \sigma\,dW_t$$ Compute the fair price of a Cash or Nothing Option with pay-off function $$V(S_T) = \mathbb{1}_{S_T.

Hint: Replace $$\mu$$ such that, the discounted price at maturity $$S(T)$$ under the risk-free measure is a martingale.

It means that the option can just be exercised at maturity $$T$$ and has value $$1$$ when at maturity the underlying price is below the strike.

My thoughts: Use a discretization process like Euler-Maruyama and then compute recursively the value of $$S(T)$$. Then using the pay-off function, approximate it with a Monte-Carlo simulation.

However, I don't know how to use this Hint. My professor said it could be really useful but I do not know how to use it. Any help with this problem would be really meaningful.

Many thanks.

Set $$\mu=r-q$$ (if you have dividends, or simply $$\mu=r$$ if there are no dividends). So if you change from the real worl probability measure $$\mathbb{P}$$ to the risk-neutral measure $$\mathbb{Q}$$ you get that $$\mathrm{d}S_t=(r-q)\mathrm{d}t+\sigma \mathrm{d}W_t$$. Then, using risk-neutral pricing, the inital value of your claim is given by \begin{align*} V_0 &= e^{-rT} \mathbb{E}^\mathbb{Q}[{1}_{\{S_T< K\}}] \\ &= e^{-rT} \mathbb{Q}[\{S_T< K\}]. \end{align*}
Thus, all you need to do is to find the probability distribution of $$S_T$$ under $$\mathbb{Q}$$. Using again that $$\mathrm{d}S_t=(r-q)\mathrm{d}t+\sigma \mathrm{d}W_t$$, we see that $$(S_t)$$ is an arithmetric Brownian motion under $$\mathbb{Q}$$ and thus normally distributed. Furthermore, \begin{align*} S_T= S_0+(r-q)T + \sigma W_T \sim N\big(S_0+(r-q)T,\sigma^2T\big), \end{align*} since $$W_T\sim N(0,T)$$. Now, set $$m=S_0+(r-q)T$$ and $$s=\sigma\sqrt{T}$$. Then, $$S_T=m+sZ$$ where $$Z\sim N(0,1)$$. Thus, \begin{align*} V_0 &= e^{-rT} \mathbb{Q}[\{m+sZ< K\}]\\ &= e^{-rT} \mathbb{Q}\left[\left\{Z< \frac{K-m}{s}\right\}\right]\\ &= e^{-rT} \Phi\left(\frac{K-m}{s}\right)\\ &= e^{-rT} \Phi\left(-\frac{S_0-K+(r-q)T}{\sigma\sqrt{T}}\right) \\ &= e^{-rT} \left(1- \Phi\left(\frac{S_0-K+(r-q)T}{\sigma\sqrt{T}}\right)\right), \end{align*}
where $$\Phi$$ denotes the cumulative distribution function of a standard normal distribution.
By the way, in the Black-Scholes model, the price of a Cash-Or-Nothing option is given by $$e^{-rT}\Phi(-d_2)=e^{-rT}\big(1-\Phi(d_2)\big)$$, see here.